Question

The domain of definition of the functions $y(x)$ given by the equation $a^{ x }+a^y=a(a>1)$ is

• A. $0< x \leq 1$
• B. $0 \leq x \leq 1$
• C. $-\infty< x< 1$
• D. $-\infty< x \leq 0$

Answer 1

Answer: (C)

$a^{x-1}+a^{y-1}=1 $
$\Rightarrow a^{y-1}=1-a^{x-1} \Rightarrow(y-1) \log a=\log \left(1-a^{x-1}\right) $
$\text { R.H.S is defined if } 1-a^{x-1}>0 $
$\Rightarrow a^{x-1}<1=a^{\circ} \Rightarrow x-1< 0 \text { as } a >1 $
$\Rightarrow x< 1 \text { and the required domain is }-\infty < x< 1 \text {. }$